Partition Equation

Theorem

Let group $G$ act on a finite set $X$.

Let the distinct orbits of $X$ under the action of $G$ be:

$\operatorname{Orb} \left({x_1}\right), \operatorname{Orb} \left({x_2}\right), \ldots, \operatorname{Orb} \left({x_s}\right)$

Then:

$\left|{X}\right| = \left|{\operatorname{Orb} \left({x_1}\right)}\right| + \left|{\operatorname{Orb} \left({x_2}\right)}\right| + \cdots + \left|{\operatorname{Orb} \left({x_s}\right)}\right|$

Proof

Follows trivially from the fact that the orbit of an element is an equivalence class.

$\blacksquare$